How does batch normalisation actually work?
A batch normalisation layer is like a standard FC layer but instead of learning weights and bias', you learn means and variances and scale the whole layer by said means and variances.
Fact 1:
Because it behaves just like a normal layer, and can learn, with the right structure it will learn to get a high enough accuracy.
Fact 2
Disabling learning on a batch norm layer is just like disabling learning on any other layer. It will not updated any of its parameters, and in this case the parameters are the means and variances, and so these will not be updated.